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<math>L^p</math> spaces are widely used in mathematics and applications.
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The [[Fourier transform]] for the real line (resp. for periodic functions, cf. [[Fourier series]]) maps <math>L^p(\mathbb{R})</math> to <math>L^q(\mathbb{R})</math> (resp. <math>L^p(\mathbb{T})</math> to <math>\ell^q</math>), where <math>1 \leq p \leq 2</math> and 1/''p''+1/''q''=1. This is a consequence of the [[Riesz-Thorin theorem]], and is made precise with the [[Hausdorff-Young inequality]].
By contrast, if p>2, the Fourier transform does not map into <math>L^q</math>.
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